Low-IF architectures have a number of advantages for low cost radio receivers, most notably the absence of an external filter component and insensitivity to DC offsets in the receiver chain.
A prior art example low-IF receiver structure is pictured in FIG. 1. A radio signal is initially received by an antenna 1, and a low noise amplifier (LNA) 3 provides initial amplification of the received signal. A quadrature mixer 5 then mixes the signal with the output of a local oscillator 7 into in-phase I and quadrature Q components with the frequency reduced so that the wanted signal is centered around the low-IF frequency. The signal is then filtered through a polyphase filter 9, which has a band pass frequency response centered around the wanted signal to allow only the wanted signal to pass trough. Following amplification by a programmable gain amplifier (PGA) stage 11, the signal is by means of an analog-to-digital converter 13 converted to digital form for demodulation.
An ideal frequency plan of the low-IF receiver structure is shown in FIG. 2a. The frequency Fc of the local oscillator 7 of the receiver is tuned to the frequency of the wanted signal 21 minus the low-IF frequency. The lower adjacent channel 23 is typically at a negative frequency after having been mixed in the quadrature mixer 5 as indicated in FIG. 2a. The polyphase filter 9, whose band pass frequency response is shown by the dashed curve 25, allows the wanted channel to pass through.
However, in practice, a number of effects interfere with the desired operation. The input signals from the local oscillator 7 to the mixer should ideally be sine and cosine signals, i.e. be sinusoids with a 90° phase difference. However, in practice there will be a slight phase error between the two signals. Additionally, there may be an amplitude error between the in-phase and quadrature signals caused by differences in gain between the two paths. The net result of these imperfections is that energy from negative frequencies, i.e. frequencies below the carrier frequency, are reflected or mirrored to appear as images at an equal positive frequency, as shown at 27 in FIG. 2b. 
This reflection phenomenon is a severe problem in situations where the adjacent channel image is found within the pass band of the wanted signal, since it is not attenuated by the polyphase filter and thus appears as interference. Since, for example, in a WLAN 802.11g receiver the adjacent channel signal can be up to 35 dB stronger than the wanted channel signal, this effect can be a limiting factor.
The effect of the phase and amplitude error can be well modeled as a linear transformation on the signal I, Q, as follows excluding the polyphase filter:
                              (                                                                      I                  ′                                                                                                      Q                  ′                                                              )                =                              K            ⁡                          (                                                                                          cos                      ⁢                                                                                          ⁢                      ϕ                                                                                                  sin                      ⁢                                                                                          ⁢                      ϕ                                                                                                            0                                                                              1                      +                      Δ                                                                                  )                                ⁢                      (                                                            I                                                                              Q                                                      )                                              (        1        )            where I′, Q′ are the distorted signal, φ is the phase error, Δ is the relative amplitude error, and K is a constant.
A straightforward way to eliminate the mismatch is to perform the inverse operation on the signal, using digital multiplication and summation operations:
                              (                                                    I                                                                    Q                                              )                =                              K                                          (                                  1                  +                  Δ                                )                            ⁢              cos              ⁢                                                          ⁢              ϕ                                ⁢                      (                                                                                1                    +                    Δ                                                                                                              -                      sin                                        ⁢                                                                                  ⁢                    ϕ                                                                                                0                                                                      cos                    ⁢                                                                                  ⁢                    ϕ                                                                        )                    ⁢                      (                                                                                I                    ′                                                                                                                    Q                    ′                                                                        )                                              (        2        )            
A further small gain scaling or permitted gain error gives a simplified result:
                              (                                                    I                                                                    Q                                              )                =                                            K              ′                        ⁡                          (                                                                                          1                      +                      α                                                                            β                                                                                        0                                                        1                                                              )                                ⁢                      (                                                                                I                    ′                                                                                                                    Q                    ′                                                                        )                                              (        3        )            where K′=K/(1+Δ), 1+α=(1+Δ)/cos φ, and β=−sin φ/cos φ.
There are a number of other manners to represent the error and the resulting compensation equation. However, they all have a similar effect.